gnicholl / ksmirnovk

Implementation of k-sample version Kolmogorov-Smirnov test in Stata

ksmirnovk

Implementation of k-sample version Kolmogorov-Smirnov test in Stata. Read more discussion on my blog post: https://gnicholl.github.io/post/ksmirnovk/

Required package

ssc install integrate.pkg

Description of program

• syntax: ksmirnovk y , by(x)

  • y: variable of interest
  • x: categorical variable specifying k groups numbered 1,2,...,k

• ksmirnovk produces a p-value for testing the null hypothesis that the distribution of y is the same in all k groups of x

  • It is a non-parametric test, and is the k-sample analogue of the ksmirnov command in Stata.
  • For k=2 groups, ksmirnovk is theoretically identical to ksmirnov, though I include more terms in the approximation of the p-value, so output may differ slightly
  • As a side effect, it also produces a plot of the empirical cdfs of each group

• I compute the test based on Kiefer (1959)

  • the test statistic requires the computation of each group's empirical cdf, as well as the empirical cdf of the pooled data. The statistic is based on a sum of max (squared) differences between the group and pooled cdfs.
  • To compute a p-value, we need to calculate the cdf of the test statistic
    • Given k groups,
      • if k=2 (standard ksmirnov case) or k=4, there is an easy, closed-form expression
      • otherwise, it is a numercially intensive procedure.
        • the function is an infinite series, and each term depends on finding the root of a bessel function, and evaluating a different bessel function at that root
        • bessel functions themselves do not have a closed form solution, and instead can be represented by an infinite sum or a difference of two integrals. Here I use the integral representation (see Temme 1996) and use 200 quadrature points
      • To save time, I pre-compute the first 30 roots of the bessel function for cases k = 1,3,5,6,7,8,9,10. I found that 30 roots was enough to reproduce Kiefer (1959)'s results (which go to 6 decimal places) (see LookupTable.xlsx).
      • for k>10, the program has to compute the bessel roots on the fly--not recommended, it is slow and has NOT been tested

Sources

• K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-V. Mises Test
  • by J. Kiefer
  • The Annals of Mathematical Statistics, Jun 1959, Vol 30, No 2 pp. 420-447
  • https://www.jstor.org/stable/2237091
• Special Functions: An introduction to the classical functions of mathematical physics
  • by Temme, Nico M. (1996) pp. 228-231
• chebroots adapted from matlab code here: https://github.com/chebfun/chebfun